The Annals of Applied Probability

Improved Fréchet–Hoeffding bounds on $d$-copulas and applications in model-free finance

Thibaut Lux and Antonis Papapantoleon

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We derive upper and lower bounds on the expectation of $f(\mathbf{S})$ under dependence uncertainty, that is, when the marginal distributions of the random vector $\mathbf{S}=(S_{1},\ldots,S_{d})$ are known but their dependence structure is partially unknown. We solve the problem by providing improved Fréchet–Hoeffding bounds on the copula of $\mathbf{S}$ that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of $[0,1]^{d}$, the value of a functional of the copula is prescribed or different types of information are available on the lower dimensional marginals of the copula. We then show that, in contrast to the two-dimensional case, the bounds are quasi-copulas but fail to be copulas if $d>2$. Thus, in order to translate the improved Fréchet–Hoeffding bounds into bounds on the expectation of $f(\mathbf{S})$, we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators. By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved Fréchet–Hoeffding bounds to bounds on the expectation of $f(\mathbf{S})$. Finally, we apply these results to compute model-free bounds on the prices of multi-asset options that take partial information on the dependence structure into account, such as correlations or market prices of other traded derivatives. The numerical results show that the additional information leads to a significant improvement of the option price bounds compared to the situation where only the marginal distributions are known.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3633-3671.

Dates
Received: March 2016
Revised: January 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328710

Digital Object Identifier
doi:10.1214/17-AAP1292

Mathematical Reviews number (MathSciNet)
MR3737934

Zentralblatt MATH identifier
06848275

Subjects
Primary: 62H05: Characterization and structure theory 60E15: Inequalities; stochastic orderings 91G20: Derivative securities

Keywords
Improved Fréchet–Hoeffding bounds quasi-copulas stochastic dominance for quasi-copulas model-free option pricing

Citation

Lux, Thibaut; Papapantoleon, Antonis. Improved Fréchet–Hoeffding bounds on $d$-copulas and applications in model-free finance. Ann. Appl. Probab. 27 (2017), no. 6, 3633--3671. doi:10.1214/17-AAP1292. https://projecteuclid.org/euclid.aoap/1513328710


Export citation

References

  • [1] Bernard, C., Jiang, X. and Vanduffel, S. (2012). A note on ‘Improved Fréchet bounds and model-free pricing of multi-asset options’ by Tankov (2011). J. Appl. Probab. 49 866–875.
  • [2] Breeden, D. and Litzenberger, R. (1978). Prices of state-contingent claims implicit in options prices. J. Bus. 51 621–651.
  • [3] Carlier, G. (2003). On a class of multidimensional optimal transportation problems. J. Convex Anal. 10 517–529.
  • [4] Chen, X., Deelstra, G., Dhaene, J. and Vanmaele, M. (2008). Static super-replicating strategies for a class of exotic options. Insurance Math. Econom. 42 1067–1085.
  • [5] d’Aspremont, A. and El Ghaoui, L. (2006). Static arbitrage bounds on basket option prices. Math. Program. 106 467–489.
  • [6] Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2002). The concept of comonotonicity in actuarial science and finance: Theory. Insurance Math. Econom. 31 3–33.
  • [7] Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2002). The concept of comonotonicity in actuarial science and finance: Applications. Insurance Math. Econom. 31 133–161.
  • [8] Gaffke, N. Maß- und Integrationstheorie. Lecture Notes, Univ. Magdeburg. Available at: http://www.ku-eichstaett.de/fileadmin/150105/Masstheorie.pdf.
  • [9] Genest, C., Quesada-Molina, J. J., Rodríguez-Lallena, J. A. and Sempi, C. (1999). A characterization of quasi-copulas. J. Multivariate Anal. 69 193–205.
  • [10] Georges, P., Lamy, A.-G., Nicolas, E., Quibel, G. and Roncalli, T. (2001). Multivariate survival modelling: A unified approach with copulas. Preprint, SSRN:1032559.
  • [11] Hobson, D., Laurence, P. and Wang, T.-H. (2005). Static-arbitrage upper bounds for the prices of basket options. Quant. Finance 5 329–342.
  • [12] Hobson, D., Laurence, P. and Wang, T.-H. (2005). Static-arbitrage optimal subreplicating strategies for basket options. Insurance Math. Econom. 37 553–572.
  • [13] Lux, T. (2017). Model uncertainty, improved Fréchet–Hoeffding bounds and applications in mathematical finance Ph.D. thesis, TU Berlin.
  • [14] Makarov, G. D. (1981). Estimates for the distribution function of the sum of two random variables with given marginal distributions. Teor. Veroyatn. Primen. 26 815–817.
  • [15] Mardani-Fard, H. A., Sadooghi-Alvandi, S. M. and Shishebor, Z. (2010). Bounds on bivariate distribution functions with given margins and known values at several points. Comm. Statist. Theory Methods 39 3596–3621.
  • [16] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley, Chichester.
  • [17] Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed. Springer, New York.
  • [18] Nelsen, R. B., Quesada-Molina, J. J., Rodríguez-Lallena, J. A. and Úbeda-Flores, M. (2001). Bounds on bivariate distribution functions with given margins and measures of association. Comm. Statist. Theory Methods 30 1155–1162.
  • [19] Peña, J., Vera, J. C. and Zuluaga, L. F. (2010). Static-arbitrage lower bounds on the prices of basket options via linear programming. Quant. Finance 10 819–827.
  • [20] Preischl, M. (2016). Bounds on integrals with respect to multivariate copulas. Depend. Model. 4 277–287.
  • [21] Rachev, S. T. and Rüschendorf, L. (1994). Solution of some transportation problems with relaxed or additional constraints. SIAM J. Control Optim. 32 673–689.
  • [22] Rodríguez-Lallena, J. A. and Úbeda-Flores, M. (2004). Best-possible bounds on sets of multivariate distribution functions. Comm. Statist. Theory Methods 33 805–820.
  • [23] Rodríguez-Lallena, J. A. and Úbeda-Flores, M. (2009). Some new characterizations and properties of quasi-copulas. Fuzzy Sets and Systems 160 717–725.
  • [24] Rüschendorf, L. (1980). Inequalities for the expectation of $\Delta$-monotone functions. Z. Wahrsch. Verw. Gebiete 54 341–349.
  • [25] Rüschendorf, L. (1981). Sharpness of Fréchet-bounds. Z. Wahrsch. Verw. Gebiete 57 293–302.
  • [26] Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process. J. Statist. Plann. Inference 139 3921–3927.
  • [27] Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229–231.
  • [28] Tankov, P. (2011). Improved Fréchet bounds and model-free pricing of multi-asset options. J. Appl. Probab. 48 389–403.
  • [29] Taylor, M. D. (2007). Multivariate measures of concordance. Ann. Inst. Statist. Math. 59 789–806.
  • [30] Tsetlin, I. and Winkler, R. L. (2009). Multiattribute utility satisfying a preference for combining good with bad. Manage. Sci. 55 1942–1952.